Polymerization

Typically, in a solvent, monomer molecules combine together in the presence of an initiator together in a chemical reaction to form polymer chains or 3D networks. Polymerization involves many reaction steps such as initiation, propagation, branching, termination. Many radicals of different chain lengths are formed. The final product is a mixture containing polymer molecules of varying lengths and structure.

This feature is based on the Method of Moments Model for polymerization. Primary properties like chain length distribution of polymer molecules are obtained by classical method of moments.

Simcenter STAR-CCM+ solves for the scalar transport of three Live polymers, ${\lambda }_0$ , ${\lambda }_1$ , and ${\lambda }_2$ , and three Dead polymers, ${\mu }_0$ , ${\mu }_1$ , and ${\mu }_2$ . In addition to the source terms described below, Simcenter STAR-CCM+ allows you to add moment sources that you define. Simcenter STAR-CCM+ also solves for the scalar transport of the Initiator, Monomer, Radical, Solvent, and Modifier.

The source terms of the moment transport equations depend on the sub processes of polymerization:

• Initiator Decomposition

  $$I_i\stackrel{k_{di}}{\to }2R^{\cdot },i=1,2,\mathrm{...},N_i$$

  (3781)
• Chain Initiation

  $$R^{\text{.}}+M\stackrel{k_I}{\to }{R}_1$$

  (3782)

  where,

  the initiator consumption rate is found using:

  $$r_{Ii}=-k_{di}\left[I_i\right],i=1$$

  (3783)

  and the primary radical formation rate is found using:

  $$r_{R^{\text{.}}}={\displaystyle \sum _{i=1}^{N_d}}2f_i{k}_{di}\left[I_i\right]-k_I\left[R^{\text{.}}\right]\left[M\right]$$

  (3784)

  where,

  $f_i$ = initiator efficiency

• Propagation

  $$R_x+M\stackrel{k_p}{\to }{R}_{x+1}$$

  (3785)

  where the monomer consumption rate is found using:

  $$r_M=-k_p\left[M\right]{\lambda }_0$$

  (3786)
• Chain transfer to monomer

  $$R_x+M\stackrel{k_{tm}}{\to }{P}_x+R_1$$

  (3787)
• Chain transfer to solvent

  $$R_x+S\stackrel{k_{ts}}{\to }{P}_x+R_1$$

  (3788)

  where the solvent consumption rate is found using:

  $$r_S=-k_{ts}\left[S\right]{\lambda }_0$$

  (3789)
• Chain transfer to polymer

  $$R_x+P_y\stackrel{k_{ts}}{\to }{R}_y+P_x$$

  (3790)
• $\beta$ -Scission

  $$R_x+P_y\stackrel{k_{\beta }}{\to }{P}_x+R_z+P_{y-z}^{=}$$

  (3791)
• Reaction with terminal double bond

  $$R_x+{P_y}^{\text{=}}\stackrel{k_{db}}{\to }{R}_{x+y}$$

  (3792)
• Termination by disproportionation

  $$R_x+R_y\stackrel{k_{td}}{\to }{P}_x+P_y$$

  (3793)
• Termination by combination

  $$R_x+R_y\stackrel{k_{tc}}{\to }{P}_{x+y^{\text{.}}}$$

  (3794)

Simcenter STAR-CCM+ solves scalar transport equations for moments as follows:

• For Live Polymers

  ${\lambda }_0$ is solved using $r_{{\lambda }_0}$ as the source term

  $${\lambda }_0={\displaystyle \sum _{n=1}^{\infty }}{R}_n$$

  (3795)

  where $r_{{\lambda }_0}$ is defined by:

${\lambda }_1$ is solved using $r_{{\lambda }_1}$ as the source term

where $r_{{\lambda }_1}$ is defined by:

${\lambda }_2$ is solved using $r_{{\lambda }_2}$ as the source term

where $r_{{\lambda }_2}$ is defined by:

• For Dead Polymers

  ${\mu }_0$ is solved using $r_{{\mu }_0}$ as the source term

  $${\mu }_0={\displaystyle \sum _{n=1}^{\infty }}{P}_n$$

  (3801)

  where $r_{{\mu }_0}$ is defined by:

  $$r_{{\mu }_0}=(k_{tm}\left[M\right]+k_{ts}\left[S\right]){\lambda }_0+(\frac{1}{2}{k}_{tc}+k_{td})\lambda \begin{array}{c}2\\ 0\end{array}-k_{db}{\lambda }_0{\mu }_0+k_{\beta }{\lambda }_0{\mu }_1$$

  (3802)

${\mu }_1$ is solved using $r_{{\mu }_1}$ as the source term

where $r_{{\mu }_1}$ is defined by:

${\mu }_2$ is solved using $r_{{\mu }_2}$ as the source term

where $r_{{\mu }_2}$ is defined by:

The closure of the higher moment, ${\mu }_3$ , is adopted from Pladis and Kiparissides [779].

In addition to the source terms and their jacobians for polymer scalars and polymer moments that are mentioned above, you can also include user-defined rates and jacobians for polymer scalar and moments sources.

Based on the specified Heat of Propagation and Heat of Initiation, appropriate source terms are added to the energy equation.

By solving the transport of the above six Moment equations, you can obtain information about the polymer molecular structure, such as Polydispersity and the Molecular Weight Distribution of the polymer. Simcenter STAR-CCM+ provides post-processing field functions for Polydispersity Index, Polymer Number-Averaged, and Mass-Averaged Chain Lengths. All Live and Dead Moments are also available as post-processing field functions.

Polydispersity Index is given by:

Polymer Number-Averaged Chain Length (NACL) is given by:

Polymer Mass-Averaged Chain Length (MACL) is given by: